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Journal of Algebra Applications of mapping cones over Clements–Lindström rings
Applications of mapping cones over Clements–Lindström rings
Vesselin Gasharov, Satoshi Murai, Irena PeevaBu kitabı nə dərəcədə bəyəndiniz?
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325
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2011
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10.1016/j.jalgebra.2010.10.006
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Journal of Algebra 325 (2011) 34–55 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra Applications of mapping cones over Clements–Lindström rings Vesselin Gasharov a , Satoshi Murai b , Irena Peeva a,∗ a b Department of Mathematics, Cornell University, Ithaca, NY 14853, USA Department of Mathematical Science, Faculty of Science, Yamaguchi, University, 16771 Yoshida, Yamaguchi 7538512, Japan a r t i c l e i n f o Article history: Received 24 July 2009 Available online 27 October 2010 Communicated by Steven Dale Cutkosky a b s t r a c t We prove that Gotzmann’s Persistence Theorem holds over every Clements–Lindström ring. We also construct the inﬁnite minimal free resolution of a squarefree Borel ideal over such a ring. © 2010 Elsevier Inc. All rights reserved. MSC: 13D02 Keywords: Free resolutions Betti numbers Hilbert functions 1. Introduction a a Throughout the paper S = k[x1 , . . . , xn ] is a polynomial ring over a ﬁeld k. Let P = (x11 , . . . , xnn ), = 0) and set W = S / P . We say that W is a Clements–Lindström with a1 a2 · · · an ∞ (here x∞ i ring. In this paper, we work over the quotient ring W . The wellknown Clements–Lindström Theorem [CL] states that for every graded ideal J in W there exists a lex ideal with the same Hilbert function; this is a generalization of Macaulay’s Theorem [Ma] to W . Green’s Theorem in S, cf. [Gr], is generalized to W by Mermin and Peeva in [MP]. Bigatti, Hulett, and Pardue proved that a lex ideals in S attains the maximal possible Betti numbers among all ideals with the same Hilbert function; we generalized this to W in [MP2]. It was an open question to generalize Gotzmann’s Persistence Theorem [Go] to W ; see Problem 5.6 in [PS2]. It is proved by Gasharov [Ga] that Gotzmann’s Persistence Theorem holds over W for all Borel ideals. We show in * Corresponding author. Email address: irena@math.cornell.edu (I. Peeva). 00218693/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016; /j.jalgebra.2010.10.006 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 35 Theorem 3.9 that Gotzmann’s Persistence Theorem holds over W for all graded ideals. Our proof also provides a new proof of Gotzmann’s Persistence Theorem over the polynomial ring S. A lot is known about the properties of Borel ideals and squarefree Borel ideals in the polynomial ring S. The minimal free resolution of a Borel ideal over S is the wellknown Eliahou–Kervaire resolution [EK]; by [AHH2] the minimal free resolution of a squarefree Borel ideal over S is also given by the Eliahou–Kervaire construction. The minimal free resolution of a squarefree Borel ideal over an exterior algebra is constructed in [AAH]. In Section 4, we construct the minimal free resolution of a squarefree Borel ideal over the Clements–Lindström ring W . The proof of Theorem 4.11 shows that the resolution is obtained using repeatedly mapping cones and Tate’s minimal free resolution [Ta]. It implies Corollary 4.15, which provides a formula for the Betti numbers. In Section 3 we consider Borel ideals in W . Using repeatedly mapping cones and Tate’s minimal free resolution [Ta] we obtain a (possibly nonminimal) free resolution of a Borel ideal; this leads to a coeﬃcientwise upper bound for its Poincarè series. Aramova, Herzog and Hibi [AHH,AHH3] proved that squarefree lex ideals have the largest graded Betti numbers among all squarefree monomial ideals for a ﬁxed Hilbert function over a polynomial ring as well as over an exterior algebra. In Section 5, we generalize this result to Clements–Lindström rings of the form W = S /(xa1 , xa2 , . . . , xna ). 2. Preliminaries Throughout this section, M is a monomial ideal in the Clements–Lindström ring W . Let m1 , . . . , mr be the minimal set of monomial generators of M. First, we recall how to multigrade W and the minimal free resolution of W / M. The polynomial ring S is Nn graded by setting mdeg(xi ) to be the i’th standard vector in Nn . We say that S is multigraded instead of Nn graded, and we say multidegree instead of Nn degree. For every (h1 , . . . , hn ) ∈ Nn h h there exists a unique monomial m of that degree, namely x11 · · · xnn . Every monomial ideal in S is multigraded. Therefore, the Clements–Lindström ring W and the monomial ideal M are multigraded. We say that a monomial m ∈ S is P free if its image in the quotient ring W is nonzero, that is, a for each 1 i n we have that xi i does not divide m. If m ∈ S is a P free monomial, then denote by W (−m) the free W module with one generator in multidegree m. Since the monomial ideal M is multigraded, there exists a minimal free resolution of W / M over W which is multigraded. Therefore, we have multigraded Betti numbers which we denote as follows W b iW ,m ( W / M ) = dimk Tori ,m ( W / M , k) for i 0, m a monomial. Therefore, the multigraded minimal free resolution of W / M over W can be written as ··· → m W b3,m (−m) → W b2,m (−m) → W (−ms ) → W , 1sr m where the sums run over all monomials. W We denote by b iW , j ( W / M ) = dimk Tori ( W / M , k) j the graded Betti numbers of W / M over W , and by b iW ( W / M ) the total Betti numbers. The graded Poincarè series of W / M is P W W / M (t , u ) = W W W i j b ( W / M ) t u . Furthermore, the Poincarè series of W / M is P ( t ) = b ( W / M )t i . i, j i, j i, j i W /M We are going to use a result of Tate [Ta] which provides the minimal free resolution of c c W /(x11 , . . . , xnn ) over W (here c 1 , . . . , cn 0 are natural numbers such that c i ai for each i). The graded Poincarè series of that resolution is 1 p n c p =a p (1 + tu c p ) . (1 − t 2 ua p ) (2.1) 36 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 Recall that the rate of W / M over W is rate W ( W / M ) = sup i2 , t i − 1 i−1 where t i = max j b iW , j ( W / M ) = 0 or j = i . 3. Borel ideals and Gotzmann’s Persistence Theorem in Clements–Lindström rings In this section, we provide an upper bound for the Poincarè series of a Borel ideal in a Clements– Lindström ring, and we prove that Gotzmann’s Persistence Theorem holds in all Clements–Lindström rings. An ideal M in W is called Borel if it is generated by monomials and the following property holds: if m is a P free monomial in M, xi is a variable that divides m, and 1 j i, then we have that x jm ∈ M. x i Lemma 3.1. Denote by the partial order on the monomials in W deﬁned by α α β α β α β β if deg x1 1 · · · xn n = deg x1 1 · · · xn n x1 1 · · · xn n x1 1 · · · xn n and αi < βi for the last index i with αi = βi . Suppose that the minimal monomial generators m1 , . . . , mr of a Borel monomial ideal M are ordered so that i < j if either the degree of mi is less than the degree of m j , or the degrees are equal and mi m j . For 1 s r, set ms  = max{ j  x j divides ms }. For s 2 we have that (m1 , . . . , ms−1 ) : ms = if ams  = ∞, (x1 , . . . , xms −1 ) c ms  (x1 , . . . , xms −1 , xms  ) if ams  < ∞, c m  where c ms  is the minimal power so that xmss ms = 0. Proof. Consider the monomial ideal M̃ that is the smallest Borel ideal in the polynomial ring S containing the monomials m1 , . . . , mr . Let m̃1 , . . . , m̃ w be its minimal monomial generators. Order them so that i < j if either the degree of m̃i is less than the degree of m̃ j , or the degrees are equal and m̃i rlex m̃ j . Let u be such that ms = m̃u . By [PS, Lemma 2.4], we have that (m̃1 , . . . , m̃u −1 ) : m̃u = (x1 , . . . , xmu −1 ). Therefore, in S we have that (m̃1 , . . . , m̃u −1 ) + xa11 , . . . , xnan This implies the desired lemma. : m̃u = (x1 , . . . , xms −1 ) if ams  = ∞, c ms  (x1 , . . . , xms −1 , xms  ) if ams  < ∞. 2 Recall the deﬁnitions of the Poincarè series and rate in Section 2. Proposition 3.2. Set v = max{ j  a j < ∞}. Let M be a Borel ideal in W . Let m1 , . . . , mr be the minimal monomial generators of M. For 1 s r, set ms  = max{ j  x j divides ms }, and set c ms  to be the minimal c m  power so that xmss ms = 0. V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 37 (1) The graded Poincarè series P W W / M (t , u ) of W / M is coeﬃcientwise less or equal to tu deg(ms ) 1+ 1sr ms  v (1 + tu )ms −1 (1 + tu cms  ) + 2 aj 1 j ms  (1 − t u ) tu deg(ms ) 1sr ms > v (1 + tu )ms −1 . 2 aj 1 j v (1 − t u ) (2) The Poincarè series of W / M satisﬁes the coeﬃcientwise inequality t PW W / M (t ) 1 + 1sr ms  v (1 − t )ms  t (1 + t )ms −1− v + 1sr ms > v (1 − t ) v . (3) Suppose that 2 a1 · · · an < ∞. Then rate W ( W / M ) max deg(ms ) + c ms  − 1, deg(ms ) − 1 + ams  2 1sr . In the proof, we will construct a nonminimal free resolution of W / M over W , which is of interest on its own. Proof of Proposition 3.2. For a monomial m denote m = max{ j  x j divides m}, and let deg(m) be the degree of m with respect to the standard grading of S. Let E be the exterior algebra over k on basis e 1 , . . . , en . A monomial e = e p 1 ∧ · · · ∧ e pq in E is a product of some of the variables; we assume that p 1 < · · · < p q . Set e  = p q . Furthermore, let γ = (γ1 , . . . , γn ) ∈ Nn . (Throughout, for simplicity of notation, we assume that N contains 0.) Set γ  = max{ j  γ j = 0}. We denote by m1 , . . . , mr the minimal monomial generators of the ideal M. We consider the symbol {ms , e , γ }, where ms is one of the minimal monomial generators of the ideal M, e is a monomial in E, and γ ∈ Nn . Let v = max{ j  a j = ∞}. We say that {ms , e , γ } is an admissible symbol if if ms  v , ms  ms  − 1 if ms  > v , γ  min v , ms  . e  For an admissible symbol {ms , e , γ } we deﬁne its homological degree, degree, and multidegree to be hdeg {ms , e , γ } = 1 + q + 2(γ1 + · · · + γn ), c m  a γ i i ) if pq = ms , 1i min{ v ,ms } xi a i γi ms ( 1 j q x p j )( 1i min{ v ,ms } xi ) otherwise, deg(ms ) + c ms  + (q − 1) + 1i min{ v ,ms } ai γi if p q = ms , deg {ms , e , γ } = deg(ms ) + q + 1i min{ v ,ms } ai γi otherwise. mdeg {ms , e , γ } = ms xmss ( 1 j q−1 x p j )( (3.3) We are going to prove that there exists a multigraded free resolution F M of W / M that has the admissible symbols as a basis. Order m1 , . . . , mr as in Lemma 3.1. The proof is by induction on the number of minimal monomial generators of the Borel monomial ideal. Let s 1. Set J = (m1 , . . . , ms−1 ) (if s = 1, then we set J = 0). Note that J is a Borel ideal. By induction hypothesis, there exists a free resolution F J of W / J that 38 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 has the desired basis. Set N = (m1 , . . . , ms ). In order to construct a free resolution of W / N we are going to use the mapping cone associated to the short exact sequence ms 0 → W /( J : ms )(−ms ) −→ W / J → W /(m1 , . . . , ms ) = W / N → 0 (here the multidegree of the ﬁrst module W /( J : m) is shifted in order to make the sequence multihomogeneous). Let K be the minimal free resolution of the module W /( J : ms )(−ms ). By Lemma 3.1, we have that ( J : ms ) = if ms  > v , (x1 , . . . , xms −1 ) c (x1 , . . . , xms −1 , xmmss ) if ms  v . Therefore, we can obtain K following Tate’s construction [Ta]. We need to multigrade K so that the free module in homological degree 0 is W (−ms ) with one generator of multidegree ms . We denote the basis of K by the admissible symbols {ms , e , γ }; the element {ms , 0, 0} is the basis in homological degree 0. The resolution is graded and multigraded as in (3.3). There exists a comparison map −μ : K → F J such that • −μ is a map of complexes, ms • −μ lifts the homomorphism W /( J : ms )(−ms ) −−→ W/ J, • −μ is multihomogeneous. The comparison map −μ : K → F J yields a mapping cone, which is F N with the desired basis. We have shown by induction that there exists a multigraded free resolution F M of W / M with basis the admissible symbols. Next, we are going to use the resolution F M in order to obtain bounds on the Poincarè series. Fix a 1 s r. The admissible symbols of the form {ms , e , γ } (here e and γ vary) contribute to the graded Poincarè series of F M the summand tu deg(ms ) (1 + tu )ms −1 (1 + tu cms  ) 2 aj 1 j ms  (1 − t u ) if ms  v , (1 + tu )ms −1 2 aj 1 j v (1 − t u ) otherwise, tu deg(ms ) where the numerator of the fraction comes from e and the denominator comes from γ . In order to obtain the entire graded Poincarè series, we sum over all minimal monomial generators m1 , . . . , mr of M. This leads to the following formula for the graded Poincarè series of the resolution F M : tu deg(ms ) 1+ 1sr ms  v (1 + tu )ms −1 (1 + tu cms  ) + 2 aj 1 j ms  (1 − t u ) (1 + tu )ms −1 . 2 aj 1 j v (1 − t u ) tu deg(ms ) 1sr ms > v Since the resolution F M is possibly nonminimal, the above formula is an upper bound for the graded Poincarè series P W W / M (t , u ). The Poincarè series of the resolution F M is obtained from the graded Poincarè series by setting u = 1. We get 1+ t 1sr ms  v (1 + t )ms  + (1 − t 2 )ms  t 1sr ms > v (1 + t )ms −1 =1+ (1 − t 2 ) v t 1sr ms  v (1 − t )ms  t (1 + t )ms −1− v + 1sr ms > v (1 − t ) v . V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 39 Since the resolution F M is possibly nonminimal, the above formula is an upper bound for the Poincarè series P W W / M (t ). (3) Now, we consider the rate of W / M over W . Assume that 2 a1 · · · an < ∞. By (3.3), it follows that rate W ( W / M ) = max −1 + deg(ms ) + c ms  + (q − 1) + γ −1 + deg(ms ) + 1ims  ai γi , 2(γ1 + · · · + γms  ) 1i ms  ai i q + 2(γ1 + · · · + γms  ) −1 + deg(ms ) + c ms  + (q − 1) + ams  γms  −1 + deg(ms ) + ams  γms  , q + 2γms  2γms  −1 + deg(ms ) + c ms  + ams  γms  −1 + deg(ms ) + ams  γms  , = max 1 + 2γms  2γms  a (deg(ms ) + c ms  − 1) − m2 s  ams  deg(ms ) − 1 ams  = max + , + 1 + 2γms  2 2γms  2 deg(ms ) − 1 + ams  . 2 = maxs deg(ms ) + c ms  − 1, = max 2 Remark 3.4. Assume that an < ∞. Note that another (possibly nonminimal) free resolution of W / M is constructed in [GHP] (by mistake that resolution is called “minimal” in the paper). That nonminimal free resolution yields an upper bound, which can be compared to the upper bound in Proposition 3.2(1)(2). Let τ = (τ1 , . . . , τn ) ∈ Nn , supp(τ ) = {i  τi = 0}, and supp(ms ) = {i  xi divides ms }. By [GHP, Corollary 2.11] we get the following coeﬃcientwise inequality for the Poincarè series P W W / M (t , u ) PW W / M (t , u ) 1 + b Sj,τ ( S / M ) j 1 τ ∈N a −τ i (1 + tu i i a i ∈supp(τ ) ) (1 − t 2 u i i ) uτ t j . For the Poincarè series P W W / M (t ) we get the following two coeﬃcientwise inequalities; the former holds by [GHP, Corollary 2.11] and the latter holds by Proposition 3.2(2): PW W / M (t ) 1 + b Sj,τ ( S / M ) j 1 τ ∈N tj (1 − t )supp(τ ) t = 1+ 1sr (1 − t )supp(ms ) t PW W / M (t ) 1 + 1sr (1 − t )ms  + j 1 τ ∈N b Sj+1,τ ( S / M ) t j +1 (1 − t )supp(τ ) , . We will give a simple example. Take M = (x31 , x21 x2 , x21 x3 ) in W = k[x1 , x2 , x3 ]/(x51 , x52 , x53 ). For the Poincarè series P W W / M (t ) we get the following two coeﬃcientwise inequalities; the former holds by [GHP, Corollary 2.11] and the latter holds by Proposition 3.2(2): 40 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 PW W / M (t ) 1 + t + (1 − t )2 PW W / M (t ) 1 + t (1 − t ) + (1 − t ) + t + t (1 − t )2 t2 + (1 − t )2 t (1 − t )2 + (1 − t )2 t2 (1 − t )3 t + t2 (1 − t )3 + , (1 − t )3 t3 . We will show that the new bound in Proposition 3.2(2) is always better than the bound in [GHP, Corollary 2.11]. Let z, y ∈ N and z y. Consider the polynomial ring on variables u 1 , . . . , u y . Set U = k[u 1 , . . . , u z ]. We have the following equality of vector spaces k [u 1 , . . . , u y ] = U u z+i 1 · · · u z+i j U [u z+i 1 · · · u z+i j ]. 1i 1 <···<i j y − z 1 j y − z 1 Since the Hilbert series of a polynomial ring is (1−t )number of variables implies the identity 1 (1 − t ) y = 1 (1 − t )z + y−z , the above equality of vector spaces (1 − t )z+ j j 1 j y − z tj . Now, ﬁx a 1 s r. Let z be the number of different variables that divide ms (that is, z = supp(ms )), and let y = ms . The above equality yields t = (1 − t )ms  t + (1 − t )supp(ms ) ms  − supp(ms ) t j +1 j (1 − t )supp(ms )+ j 1 j ms −supp(ms ) . The minimal free resolution of S / M over the polynomial ring S is the Eliahou–Kervaire resolution [EK]. If z < y, then for every sequence 1 i 1 < · · · < i j < y of numbers that are not in supp(ms ), we have a syzygy (ms ; i 1 , . . . , i j ) of homological degree j + 1 and multidegree with support supp(ms ) ∪ xi 1 · · · xi j . Such a syzygy contributes sum j 1 ms −supp(ms ) j t j +1 S τ ∈N b j +1,τ ( S / M ) (1−t )supp(τ ) . t j +1 j (1−t )supp(ms )+ j t (1 − t )ms  = = inside the sum t (1 − t )supp(ms ) t (1 − t )supp(ms ) ms −supp(ms ) There exist j 1 j j 1 τ ∈N Therefore, ms  − supp(ms ) t j +1 j (1 − t )supp(ms )+ j 1 j ms −supp(ms ) b̃ j +1,τ inside the such syzygies, so we get t j +1 S τ ∈N b j +1,τ ( S / M ) (1−t )supp(τ ) . + + t j +1 (1−t )supp(ms )+ j t j +1 (1 − t )supp(τ ) , where the numbers b̃ j ,τ (that appear in the last sum above) count the special type of syzygies described above. When we vary ms , we obtain the desired inequality t 1+ 1sr (1 − t )ms  t 1+ 1sr (1 − t )supp(ms ) + j 1 τ ∈N b Sj+1,τ ( S / M ) t j +1 (1 − t )supp(τ ) . V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 41 The inequality is usually strict, because there usually exist syzygies that are not counted in the argument above (since they don’t provide new support). Note that we have a strict inequality in the simple example given above. Next, we focus on Gotzmann’s Persistence Theorem. Order the variables by x1 > · · · > xn , and consider the lex order lex of the monomials in W . A monomial ideal L in W is called lex if the following property holds: if m ∈ L is a monomial and m lex m is a monomial of the same degree then m ∈ L. Clements–Lindström’s Theorem 3.5. (See [CL].) For every graded ideal in W there exists a lex ideal with the same Hilbert function. Gotzmann’s Persistence Theorem [Go] over the polynomial ring S states: Gotzmann’s Persistence Theorem 3.6. (See [Go].) Suppose that W = S. Let I be a graded ideal in S, and L be the lex ideal with the same Hilbert function. If q ∈ N is such that I is generated in degrees q and dimk I q+1 = dimk S 1 L q , then dimk I q+i = dimk S i L q for all i 0. In other words, L is generated in degrees q. The above result does not hold over W , as shown by the next example by Gasharov. Example 3.7. (See [Ga].) Let W = k[x1 , x2 , x3 ]/(x31 , x32 , x33 ). Let I be the ideal generated by the monomials x21 , x1 x2 , x22 . Take q = 2. Then L 2 is the vector space spanned by the monomials x21 , x1 x2 , x1 x3 . We have that dimk I 2 = dimk L 2 = 3, dimk I 3 = dimk W 1 L 2 = 5, dimk I 4 = 6 > dimk W 2 L 2 = 5. The following version of Gotzmann’s Persistence Theorem over W is proved for Borel ideals by Gasharov: Gotzmann’s Persistence Theorem 3.8. (See [Ga].) Set f = max 1, {a p − 1  1 p n, a p < ∞} . Let I be a Borel ideal in W , and L be the lex ideal with the same Hilbert function. If q ∈ N is such that I is generated in degrees q and the equalities dimk I q+i = dimk W i L q for 0 i f 42 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 hold, then dimk I q+i = dimk W i L q for all i 0. In other words, L is generated in degrees q. We prove Gotzmann’s Persistence Theorem for all graded ideals: Gotzmann’s Persistence Theorem 3.9. Set f = max 1, {a p − 1  1 p n, a p < ∞} . Let I be a graded ideal in W , and L be the lex ideal with the same Hilbert function. If q ∈ N is such that I is generated in degrees q and the equalities dimk I q+i = dimk W i L q for 0 i f dimk I q+i = dimk W i L q for all i 0. hold, then In other words, L is generated in degrees q. The proof of Gotzmann’s Persistence Theorem 3.6 over the polynomial ring S in [Gr] uses a reduction to the Borel case. The reduction is achieved by taking a generic initial ideal. This argument does not work over W because we don’t have generic changes of variables in the Clements–Lindström ring. We are not aware how to reduce the proof of Theorem 3.9 to the Borel case in Theorem 3.8. Our proof of Theorem 3.9 uses consecutive cancellations of Betti numbers from [MP2]. It provides a new proof of Gotzmann’s Persistence Theorem over S. Proof. The equalities dimk I q+i = dimk W i L q for 0 i f show that the ideal L does not have a minimal monomial generator in degrees j such that q + 1 j q + f . We will show that L does not have a minimal monomial generator in degree q + f + 1. First, we recall the deﬁnition of consecutive cancellation. Given a sequence of numbers {c i , j }, we obtain a new sequence by a cancellation as follows: ﬁx a j, and choose i and i so that one of the numbers is odd and the other is even; then replace c i , j by c i , j − 1, and replace c i , j by c i , j − 1. We have a consecutive cancellation when i = i + 1. The following result is proved in [MP2]: the graded W Betti numbers b iW , j ( W / J ) can be obtained from the graded Betti numbers b i , j ( W / L ) by a sequence of consecutive cancellations. Suppose that L has a minimal monomial generator in degree q + f + 1. Hence, we have that b1W,q+ f +1 ( W / L ) = 0. On the other hand, we know that I does not have a minimal monomial gen erator in degree q + f + 1, so b1W,q+ f +1 ( W / I ) = 0. Since b1W,q+ f +1 ( W / I ) = 0 is obtained from b1W,q+ f +1 ( W / L ) = 0 by consecutive cancellations, it follows that b2W,q+ f +1 ( W / L ) = 0. The ideal L is Borel, so we can apply 3.2. Since L does not have a minimal monomial generator in degrees j such that q + 1 j q + f , it follows that b2W,q+ f +1 ( W / L ) = 0. This is a contradiction. We proved that L does not have a minimal monomial generator in degree q + f + 1. The theorem holds by induction on degree. 2 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 43 4. Resolving squarefree Borel ideals First, we recall the deﬁnition of a squarefree Borel ideal. A monomial is squarefree if it is not divisible by the square of any of the variables. A monomial ideal is called squarefree if it can be generated by squarefree monomials. A squarefree monomial ideal M is called squarefree Borel if it satisﬁes the following squarefree Borel property: whenever i < j and g is a monomial such that gx j ∈ M and gxi is squarefree, we have gxi ∈ M as well. It is easy to see that the following holds: a squarefree monomial ideal M is squarefree Borel if and only if whenever i < j and g is a monomial such that gx j is a minimal monomial generator of M and gxi is squarefree, we have gxi ∈ M as well. Notation 4.1. Throughout this section, for simplicity we assume that an < ∞ and that char(k) = 0. We a a work over the quotient ring W = k[x1 , . . . , xn ]/(x11 , . . . , xnn ), where 2 a1 · · · an < ∞. Furthermore, M stands for a squarefree Borel monomial ideal in W . For a P free monomial m denote m = max{ j  x j divides m}, {m} = { j  x j divides m}, deg(m) = the degree of m with respect to the standard grading of S . Since M is squarefree Borel, it is easy to see that if w ∈ M is a squarefree monomial, then there is a unique decomposition w = uv, such that u is a minimal monomial generator of M and max(u ) min{ j  x j divides v }. We set b( w ) = u and call it the beginning of the monomial w. If w is a nonsquarefree monomial, then we set b( w ) = 0. Let E be the exterior algebra over k on basis e 1 , . . . , en . A monomial e in E is a product of some of the variables. We set e  = max{i  e i divides e }, {e } = {i  e i divides e }, deg(e ) = the degree of e with respect to the standard grading of E . If e = e j 1 · · · e jt and 1 j 1 < · · · < jt n, then we deﬁne σ ( j i , e ) = i − 1. Hence, e = (−1)σ ( j i ,e) e j i (e \ e j i ). Furthermore, we denote by α p the p’th standard vector in Nn . Throughout, for simplicity of notation, we assume that N contains 0. Let γ1 , . . . , γn ∈ N and γ p α p ∈ Nn . γ = (γ1 , . . . , γn ) = 1 p n Set γ  = max{ p  γ p = 0}, {γ } = { p  γ p = 0}, deg(γ ) = γ1 + · · · + γn . We call {m}, {e }, {γ } the support of m, e, γ respectively. 44 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 Deﬁnition 4.2. We use the notation in 4.1. We say that {e , γ } is a base, and denote by B the set of bases. We denote by m1 , . . . , mr the squarefree minimal monomial generators of the ideal M. For each ms and for each base {e , γ }, we consider the symbol {ms , e , γ }. We say that {ms , e , γ } is an admissible element if e  ms  and γ  ms . Otherwise, we set {ms , e , γ } = 0. We say that the base {e , γ } supports the admissible element above. We say that an admissible element is a basic admissible element if it has the form {ms , e i , 0} or {ms , 0, αi }. If {e , γ } is a base, then we deﬁne the operation •addition by {ms , e , γ } • e , γ = ms , ee , γ + γ . (4.3) Note that the multiplication in E is anticommutative, so we get ms , ee , γ = (−1)deg(e) deg(e ) ms , e e , γ , and also ee = 0 if {e } ∩ {e } = ∅. Construction 4.4. We use the notation in 4.1 and 4.2. We will construct a multigraded minimal free resolution F M of W / M over W . Let F M be the free W module on basis F = {1} ∪ {ms , e , γ } {ms , e , γ } is an admissible element, 1 s r . The module F M is graded, multigraded, and homologically graded as follows. We denote by deg, mdeg, hdeg the degree, multidegree, and homological degree, respectively. Let m = m s for some s. If {m, e , γ } is an admissible element, then we deﬁne hdeg {m, e , γ } = 1 + deg(e ) + 2 deg(γ ), mdeg {m, e , γ } = m a −1 xi i i ∈{e }∩{m} xj (ai − 1) + i ∈{e }∩{m} , p ∈{γ } j ∈{e }, j ∈{ / m} deg {m, e , γ } = deg(m) + ap γp xp 1+ j ∈{e }, j ∈{ / m} ap γp . (4.5) p ∈{γ } In particular, the following hold: the element 1 is the basis of F M in homological degree 0. The basis in homological degree 1 consists of the admissible elements {m1 , 0, 0}, . . . , {mr , 0, 0}. Next, we will deﬁne a differential on F M . Let m = ms for some s. Deﬁne a map ∂ that acts on the basic admissible elements as follows ∂ {m, 0, 0} = 0, xi {m, 0, 0} if xi ∈ / {m}, ∂ {m, e i , 0} = ai −1 xi {m, 0, 0} if xi ∈ {m}, a p −1 / {m}, ∂ {m, 0, α p } = x p {m, e p , 0} if x p ∈ x p {m, e p , 0} if x p ∈ {m}. (4.6) We deﬁne ∂ {m, e i , 0} • (−1)σ (i ,e) {e \ e i , γ } + ∂ {m, e , γ } = i ∈{e } ∂ {m, 0, α p } • {e , γ − α p }. p ∈{γ } (4.7) V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 45 Any terms that appear in (4.7) and that contain nonadmissible elements are considered to be zero. Now, we deﬁne another map μ (which comes from the comparison map in a mapping cone, as we will see later). Deﬁne a map μ that acts on the basic admissible elements as follows μ {m, 0, 0} = −m, μ {m, e i , 0} = μ {m, 0, α p } = xi m {b(xi m), 0, 0} b ( xi m ) 0 = {b(xi m), 0, 0} if xi ∈ / {m}, if xi ∈ {m}, x m − b(xpp m) {b(x p m), e p , 0} if x p ∈ / {m}, 0 = −{b(x p m), e p , 0} if x p ∈ {m}. (4.8) Furthermore, deﬁne μ {m, e i , 0} • (−1)σ (i,e) {e \ e i , γ } + μ {m, e, γ } = μ {m, 0, α p } • {e, γ − α p }. (4.9) p ∈{γ } i ∈{e } Any terms that appear in (4.9) and that contain nonadmissible elements are considered to be zero. It is clear that ∂ is multihomogeneous. We will show that μ is also multihomogeneous. Let {m, e , γ } be an admissible element. Every nonzero term of μ({m, e , γ }) either has the form xi m {b(xi m), e \ e i , γ } or has the form b(xxi imm) {b(xi m), e i ∧ e , γ − αi }. To see that the degrees of these b ( xi m ) elements are equal to that of {m, e , γ }, we must prove that {b(xi m)} ∩ {e \ e i } = {m} ∩ {e } in the former case and that {b(xi m)} ∩ {e i ∧ e } = ({m} ∩ {e }) ∪ {i } in the latter case. These follow from the fact that {m , e , γ } is admissible and {b(xi m)} = { j ∈ {m}: j i } ∪ {i } (see [AHH2, p. 361]). The differential in F M is deﬁned by d = ∂ − μ. Thus, d {m, 0, 0} = m, xm d {m, e i , 0} = d {m, 0, α p } = xi {m, 0, 0} − b(xi m) {b(xi m), 0, 0} i a −1 xi i {m, 0, 0} a p −1 xp {m, e p , 0} + x p {m, e p , 0} if xi ∈ / {m}, if xi ∈ {m}, a p −1 xp m b ( x p m) {b(x p m), e p , 0} if x p ∈ / {m}, if x p ∈ {m}, d {m, e i , 0} • (−1)σ (i ,e) {e \ e i , γ } d {m, e , γ } = i ∈{e } d {m, 0, α p } • {e , γ − α p }. + (4.10) p ∈{γ } Any terms that appear in (4.10) and that contain nonadmissible elements are considered to be zero. The differential d is homogeneous and multihomogeneous by construction. Theorem 4.11. Let 2 a2 · · · an < ∞, and char(k) = 0. Let M be a squarefree Borel ideal in W . Then F M (constructed above) is the minimal free resolution of W / M over W . We will prove the theorem after the next example. 46 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 Example 4.12. The ideal (x1 x2 , x1 x3 ) is squarefree Borel in A = k[x1 , x2 , x3 ]/(x61 , x62 , x63 ). The basis of the beginning of our resolution is: in homological degree 0: 1, in homological degree 1: {x1 x2 , 0, 0} and {x1 x3 , 0, 0}, in homological degree 2: {x1 x2 , e 1 , 0}, {x1 x2 , e 2 , 0}, {x1 x3 , e 1 , 0}, {x1 x3 , e 2 , 0}, {x1 x3 , e 3 , 0}, in homological degree 3: {x1 x2 , e 1 e 2 , 0}, {x1 x2 , 0, α1 }, {x1 x2 , 0, α2 }, {x1 x3 , e 1 e 2 , 0}, {x1 x3 , e 1 e 3 , 0}, {x1 x3 , e 2 e 3 , 0}, {x1 x3 , 0, α1 }, {x1 x3 , 0, α2 }, {x1 x3 , 0, α3 }. The beginning of the resolution is: ⎛ −x52 x1 ⎜ 5 ⎜ x1 ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎝ 0 0 0 0 0 0 0 x2 0 0 0 x3 0 −x2 x51 0 0 0 −x53 0 x51 0 0 0 −x53 x2 0 0 x1 0 0 0 x3 0 x52 0 ⎞ 0 ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎠ x3 · · · → A 9 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ 5 x1 x52 0 −x3 0 x2 x5 0 0 x51 ( x x2 x1 x3 ) A 5 −−−−−−−−−− −−−−−−−3−→ A 2 −−−1−− −−−−−→ A . The ﬁrst and longest step towards the proof of Theorem 4.11 is the following lemma. Lemma 4.13. F M is a complex. Proof. We have to show that d2 vanishes on every admissible element. Let m be one of the minimal monomial generators of M. Consider an admissible element {m, e , γ }. Throughout, ∗ stands for the unique monomial coeﬃcient that will make the expression multihomogeneous. First, we consider the basic admissible elements. Let i , p ∈ {m} and j , q ∈ / {m}. Then we have: a −1 d2 {m, e i , 0} = d xi i a −1 {m, 0, 0} = xi i m = 0, d2 {m, e j , 0} = d x j {m, 0, 0} − ∗ b(x j m), 0, 0 = x j m − ∗b(x j m) = 0, ap d2 {m, 0, α p } = d x p {m, e p , 0} = x p {m, 0, 0} = 0, {m, eq , 0} + ∗ b(xq m), eq , 0 a = xqq {m, 0, 0} − ∗ b(xq m), 0, 0 + ∗ b(xq m), 0, 0 = 0. a q −1 d2 {m, 0, αq } = d xq In the rest of the proof we assume that the admissible element {m, e , γ } is not basic. Hence, {e } ∪ {γ } has at least two elements. We consider the following cases: Case A: Let i , j ∈ {e }. We will show that d2 ({m, e , γ }) has no terms supported by the base {e \ e i e j , γ }. Case B: Suppose that γ − α p − αq 0 (coeﬃcientwise). We will show that d2 ({m, e , γ }) has no terms supported by the base {e p eq e , γ − α p − αq }. V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 47 Case C: Suppose that i ∈ {e } and γ − α p 0 (coeﬃcientwise). We will show that d2 ({m, e , γ }) has no terms supported by the base {e p e \ e i , γ − α p }. By Cases A, B, C, it follows that d2 ({m, e , γ }) = 0. Let m be a monomial. In the rest of the proof, the notation m , e , γ conditions means that the symbol is set to equal zero unless the listed conditions are satisﬁed. We call such conditions existence conditions. We will consider each of Cases A, B, C separately. Case A: Let i , j ∈ {e }. We will show that d2 ({m, e , γ }) has no terms supported by the base {e \ e i e j , γ }. Let i < j. Without loss of generality, we can assume that e = e i e j e . First, we compute (∂ − μ) {m, e , γ } = ∗{m, e \ e i , γ } − ∗{m, e \ e j , γ } − ∗ b(xi m), e \ e i , γ e \ e i  b(xi m), γ  b(xi m) + ∗ b(x j m), e \ e j , γ e \ e j  b(x j m), γ  b(x j m) + other terms. Next, we compute (∂ − μ)2 ({m, e , γ }) and label the terms that are supported by the base {e \ e i e j , γ }. We obtain that (∂ − μ)2 {m, e , γ } = ∗{m, e \ e i e j , γ } − ∗{m, e \ e i e j , γ } − ∗ b(xi m), e \ e i e j , γ e \ e i  b(xi m), γ  b(xi m) + ∗ b(x j m), e \ e i e j , γ e \ e j  b(x j m), γ  b(x j m) − ∗ b(x j m), e \ e i e j , γ e \ e i e j  b(x j m), γ  b(x j m) + ∗ b(xi m), e \ e i e j , γ e \ e i e j  b(xi m), γ  b(xi m) e \ e  b(x m) i i , γ  b ( x i x j m ) + ∗ b(xi x j m), e \ e i e j , γ e \ e i e j  b(xi x j m) e \ e  b(x m) j j , γ  b(xi x j m) − ∗ b(xi x j m), e \ e i e j , γ e \ e i e j  b(xi x j m) ( A1) ( A2) ( A3) ( A4) ( A5) ( A6) ( A7) ( A8) + terms, that are not supported by {e \ e i e j , γ }. Note that ( A1) + ( A2) = 0. We will show that ( A4) + ( A5) = 0. Note that e \ e i e j  e \ e j . We have the following three cases: (i) Suppose that b(x j m) < e \ e i e j  e \ e j . Then ( A4) = ( A5) = 0. (ii) Suppose that e \ e i e j  e \ e j  b(x j m). Then ( A4) + ( A5) = 0. 48 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 (iii) Suppose that e \ e i e j  b(x j m) < e \ e j . Then i = e \ e j . Therefore i > b(x j m) j > i, which is a contradiction. Finally, we will show that ( A3) + ( A6) + ( A7) + ( A8) = 0. Note that e \ e i e j  e \ e i . We consider the following three cases: (i) Suppose that b(xi m) < e \ e i e j  e \ e i . Then ( A3) = ( A6) = 0. Since b(xi x j m) b(xi m), it follows that ( A7) = ( A8) = 0. (ii) Suppose that e \ e i e j  e \ e i  b(xi m). Then ( A3) + ( A6) = 0. If e \ e j  b(x j m), then A (7) + ( A8) = 0. If e \ e j  > b(x j m), then e \ e j  > b(x j m) b(xi m) e \ e i  e \ e j , which is a contradiction. (iii) Suppose that e \ e i e j  b(xi m) < e \ e i . Then ( A7) = ( A3) = 0. Furthermore, it follows that j = e  and j > b(xi m). So, in (A8) we have that b(xi x j m) = b(xi m). Now ( A6) + ( A8) = 0 since e \ e j  b(xi m) b(x j m) in this case. Case B: Suppose that γ − α p − αq 0. We will show that d2 ({m, e , γ }) has no terms supported by the base {e p eq e , γ − α p − αq }. Clearly, the claim holds if either p = q, or p ∈ {e }, or q ∈ {e }. We assume that p < q and p , q ∈ / {e }. First, we compute (∂ − μ) {m, e , γ } = ∗{m, e p e , γ − α p } + ∗{m, eq e , γ − αq } + ∗ b(x p m), e p e , γ − α p e p e  b(x p m), γ − α p  b(x p m) + ∗ b(xq m), eq e , γ − αq eq e  b(xq m), γ − αq  b(xq m) + other terms. Next, we compute (∂ − μ)2 ({m, e , γ }) and label the terms that are supported by the base {e p eq e , γ − α p − αq }. We obtain that (∂ − μ)2 {m, e , γ } = − ∗ {m, e p eq e , γ − α p − αq } (B1) + ∗{m, e p eq e , γ − αq − α p } (B2) − ∗ b(x p m), e p eq e , γ − α p − αq eq e p e  b(x p m), γ − α p  b(x p m) (B3) + ∗ b(xq m), e p eq e , γ − αq − α p e p eq e  b(xq m), γ − αq  b(xq m) (B4) − ∗ b(xq m), e p eq e , γ − αq − α p e p eq e  b(xq m), γ − αq − α p  b(xq m) (B5) + ∗ b(x p m), e p eq e , γ − αq − α p e p eq e  b(x p m), γ − αq − α p  b(x p m) (B6) γ − α p  b(x p m) − ∗ b(x p xq m), e p eq e , γ − α p − αq e p eq e  b(x p xq m), (B7) γ − α p − αq  b(x p xq m) γ − αq  b(xq m) − ∗ b(x p xq m), e p eq e , γ − αq − α p e p eq e  b(x p xq m), (B8) γ − α p − αq  b(x p xq m) + terms, that are not supported by {e p eq e , γ − α p − αq }. Note that ( B1) + ( B2) = 0. V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 49 We will show that ( B4) + ( B5) = 0. Note that e p eq e  b(xq m) implies p b(xq m). Hence, if e p eq e  b(xq m) and γ − αq − α p  b(xq m), then γ − αq  b(xq m). Therefore, the existence conditions in (B4) and (B5) are equivalent. The same argument shows that ( B3) + ( B6) = 0. Note that e p eq e  b(x p m) implies q b(x p m). Hence, if e p eq e  b(x p m) and γ − αq − α p  b(x p m), then γ − α p  b(x p m). Therefore, the existence conditions in (B3) and (B6) are equivalent. Finally, we will show that ( B7) + ( B8) = 0. Note that e p eq e  b(x p xq m) implies q b(x p xq m) and p b(x p xq m). Hence, if e p eq e  b(x p xq m) and γ − αq − α p  b(x p xq m), then γ − α p  b(x p xq m) and γ − αq  b(x p xq m). Therefore, the existence conditions in (B7) and (B8) are equivalent. Case C: Suppose that i ∈ {e } and γ − α p − αq 0. We will show that d2 ({m, e , γ }) has no terms supported by the base {e p e \ e i , γ − α p }. / {e \ e i }. Without loss of generality, Clearly, the claim holds if either p ∈ {e \ e i }. Assume that p ∈ we can assume that e = e i e . First, we compute (∂ − μ) {m, e , γ } = ∗{m, e \ e i , γ } + ∗{m, e p e , γ − α p } − ∗ b(xi m), e \ e i , γ e \ e i  b(xi m), γ  b(xi m) + ∗ b(x p m), e p e , γ − α p e p e  b(x p m), γ − α p  b(x p m) + other terms. We consider the following two cases: (i) Suppose, that p = i. Then (∂ − μ) {m, e , γ } = ∗{m, e \ e p , γ } − ∗ b(x p m), e \ e p , γ e \ e p  b(x p m), γ  b(x p m) + other terms. Next, we compute (∂ − μ)2 ({m, e , γ }): (∂ − μ)2 {m, e , γ } = ∗{m, e , γ − α p } − ∗ b(x p m), e , γ − α p e \ e p  b(x p m), γ  b(x p m) + ∗ b(x p m), e , γ − α p e \ e p  b(x p m), γ  b(x p m) + terms, that are not supported by {e , γ − α p } = ∗{m, e , γ − α p } + terms, that are not supported by {e , γ − α p }. Now, note that the coeﬃcient of {m, e , γ − α p } in this expression is equal to the multidegree of ap αp , so it is x p = 0. i. We compute (∂ − μ)2 ({m, e , γ }) and label the terms that are supported (ii) Suppose, that p = by the base {e p e \ e i , γ − α p }. We obtain that 50 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 (∂ − μ)2 {m, e , γ } = ∗{m, e p e \ e i , γ − α p } (C 1) − ∗{m, e p e \ e i , γ − α p } − ∗ b(xi m), e p e \ e i , γ − α p e p e \ e i  b(xi m), γ  b(xi m) − ∗ b(x p m), e p e \ e i , γ − α p e p e  b(x p m), γ − α p  b(x p m) + ∗ b(x p m), e p e \ e i , γ − α p e p e \ e i  b(x p m), γ − α p  b(x p m) + ∗ b(xi m), e p e \ e i , γ − α p e p e \ e i  b(xi m), γ − α p  b(xi m) γ  b(xi m) − ∗ b(xi x p m), e p e \ e i , γ − α p e p e \ e i  b(xi x p m), γ − α p  b(xi x p m) e p e  b(x p m) , γ − α p  b ( x i x p m ) + ∗ b(xi x p m), e p e \ e i , γ − α p e p e \ e i  b(xi x p m) (C 2) (C 3) (C 4) (C 5) (C 6) (C 7) (C 8) + terms, that are not supported by {e \ e i e j , γ }. Note that (C 1) + (C 2) = 0. Consider (C 3) and (C 6). If e p e \ e i  b(xi m), then p b(xi m). Hence, if e p e \ e i  b(xi m), γ − α p  b(xi m) hold, then e p e \ e i  b(xi m), γ − α p  b(xi m) hold. Thus, the existence conditions in (C 3) and (C 6) are equivalent. Therefore, (C 3) + (C 6) = 0. Finally, we will show that (C 4) + (C 5) + (C 7) + (C 8) = 0. We consider the following three cases: (i) Suppose that e p e \ e i  > b(x p m). Then (C 4) = (C 5) = (C 8) = 0. Since e p e \ e i  > b(x p m) b(xi x p m), it follows that (C 7) = 0. (ii) Suppose that e p e  b(x p m). Then (C 4) + (C 5) = 0. If e p e \ e i  b(xi x p m), then p b(xi x p m). Hence, if e p e \ e i  b(xi x p m), γ − α p  b(xi x p m) hold, then γ  b(xi x p m) b(xi ). Therefore, the existence conditions in (C 7) and (C 8) are equivalent. Hence, (C 7) + (C 8) = 0. (iii) Suppose that e p e \ e i  b(x p m) < e p e . In this case, (C 4) = (C 8) = 0. The inequalities imply that i = e  > b(x p m) p. Hence, b(xi x p m) = b(x p m). Therefore, if γ − α p  b(xi x p m) b(xi m) holds, then γ  b(xi m) holds. Thus, the existence conditions in C (5) and C (7) are equivalent. Hence, (C 5) + (C 7) = 0. 2 For the proof of Theorem 4.11, we also need the following easy lemma. Lemma 4.14. Order the variables by x1 , . . . , xn and denote by rlex the revlex order in W . Let 2 a2 · · · an < ∞, and M be a squarefree Borel ideal in W . Suppose that the minimal monomial generators m1 , . . . , mr of M are ordered so that i < j if either the degree of mi is less than the degree of m j , or the degrees are equal and mi rlex m j . For s 2 we have that (m1 , . . . , ms−1 ) : ms = a −1 xi i ∈ / {ms }, 1 i ms  , xi i i ∈ {ms } . Proof of Theorem 4.11. The proof is by induction on the number of minimal monomial generators of the squarefree Borel monomial ideal. Order the minimal monomial generators m1 , . . . , mr of M as in Lemma 4.14. Let s 1. Set J = (m1 , . . . , ms−1 ) (if s = 1, then we set J = 0). Note that J is a squarefree Borel ideal. By induction hypothesis, our construction yields the minimal free resolution F J of W / J . Set N = (m1 , . . . , ms ). In order to construct the minimal free resolution of W / N we are going to use the mapping cone associated to the short exact sequence ms 0 → W /( J : ms )(−ms ) −→ W / J → W /(m1 , . . . , ms ) = W / N → 0 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 51 (here the multidegree of the ﬁrst module W /( J : ms ) is shifted in order to make the sequence multihomogeneous). Let K be the minimal free resolution of the module W /( J : ms )(−ms ). By Lemma 4.14, we have that ( J : ms ) = a −1 xi i ∈ / {ms }, 1 i ms  , xi i i ∈ {ms } . Therefore, we can obtain K following Tate’s construction [Ta]. We need to multigrade K so that the free module in homological degree 0 is W (−ms ) with one generator of multidegree ms . We denote the basis of K by the admissible symbols {ms , e , γ }; the element {ms , 0, 0} is the basis in homological degree 0. The differential in K is −∂ described in (4.6) and (4.7). The resolution is graded and multigraded by (4.5). The minimal free resolution of W / J is F J . Next, we describe a comparison map. Consider the map −μ : K → F J deﬁned by (4.8) and (4.9). Note that −μ is multihomogeneous. We will show that it is a map of complexes. We have to verify that −μ(−∂({ms , e , γ })) = d(−μ({ms , e , γ })). Indeed, μ∂ = −∂ 2 + μ∂ = (−∂ + μ)∂ = −d∂ = −d(∂ − μ + μ) = −d2 − dμ = −dμ, where we used d2 = 0 by Lemma 4.13. Furthermore, note that −μ lifts the homomorphism W /( J : ms ms ) −−→ W / J since −μ({ms , 0, 0}) = ms by (4.8). The comparison map −μ : K → F J yields a mapping cone, which is F N . We have shown by induction that F M is a free resolution of W / M. The resolution F M is minimal, since d(F N ) ⊆ (x1 , . . . , xn )F M by construction (4.10). 2 Recall the deﬁnition of Poincarè series in Section 2. Corollary 4.15. Let 2 a2 · · · an < ∞, and char(k) = 0. Let M be a squarefree Borel ideal in W . The graded Poincarè series of W / M is PW M (t , u ) =1+ tu deg(ms ) (1 + tu )ms −deg(ms ) 1 p ms  1sr i ∈{ms } (1 + tu (1 − t 2 ua p ) a i −1 ) . The Poincarè series of W / M is t PW W / M (t ) = 1 + 1sr (1 − t )ms  . Proof. Let m = ms for some s. Consider a basic admissible element of the form {m, e i , 0}. By (4.5), we have that hdeg {m, e i , 0} = 2, mdeg {m, e i , 0} = deg {m, e i , 0} = mxi a −1 mxi i if xi does not divide m, if xi divides m, deg(m) + 1 if xi does not divide m, deg(m) + ai − 1 if xi divides m. 52 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 Consider a basic admissible element of the form {m, 0, αi }. By (4.5), we have that hdeg {m, 0, α p } = 3, ap mdeg {m, 0, α p } = mx p , deg {m, 0, α p } = deg(m) + a p . We conclude that the admissible elements of the form {m, e , γ } contribute to the graded Poincarè series the summand tu deg(m) (1 + tu )m−deg(m) 1 p m i ∈{m} (1 + tu (1 − t 2 ua p ) a i −1 ) , where the numerator of the fraction comes from e and the denominator comes from γ . In order to obtain the entire graded Poincarè series, we sum over all minimal monomial generators m1 , . . . , mr of M. This leads to the following formula for the Poincarè series of W / M: 1+ tu deg(ms ) (1 + tu )ms −deg(ms ) 1 p ms  1sr i ∈{ms } (1 + tu (1 − t 2 ua p ) a i −1 ) . The Poincarè series P W W / M (t ) is obtained from the graded Poincarè series by setting u = 1. We get PW W / M (t ) (1 + t )ms −deg(ms ) t =1+ t 1sr (1 + t )ms  (1 − t 2 )ms  t =1+ 1sr i ∈{ms } (1 + t ) 2 1 p ms  (1 − t ) 1sr =1+ (1 − t )ms  . 2 Corollary 4.16. Let 2 a2 · · · an < ∞, and char(k) = 0. Let M be a squarefree Borel ideal in W . Then rate W ( W / M ) = max deg(ms ) + ams  − 2 . 1sr The rate is achieved at homological degree 2. Proof. By (4.5), for an admissible element {m, e , γ } we have that hdeg {m, e , γ } = 1 + deg(e ) + 2 deg(γ ), deg {m, e , γ } = deg(m) + (ai − 1) + i ∈{e }∩{m} 1+ j ∈{e }, j ∈{ / m} ap γp . p ∈{γ } V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 53 It follows that rate W ( W / M ) = max = max −1 + deg(ms ) + −1 + deg(ms ) + i ∈{e }∩{m} (ai − 1) + j ∈{e }, j ∈{ / m} 1 + γ 1i ms  ai i deg(e ) + 2(γ1 + · · · + γms  ) i ∈{e }∩{m} (ai − 1) + j ∈{e }, j ∈{ / m} 1 + ams  ms  γ deg(e ) + 2γms  −1 + deg(ms ) + (ams  − 1) + ams  γms  −1 + deg(ms ) + ams  γms  = max , 1 + 2γms  2γms  a (deg(ms ) + (ams  − 1) − 1) − m2 s  ams  deg(ms ) − 1 ams  = max + , + 1 + 2γms  2 2γms  2 deg(ms ) − 1 + ams  = max deg(ms ) + ams  − 2, 2 2 = max deg(ms ) + ams  − 2 . 1sr Remark 4.17. Suppose that char(k) = 0. Tate’s resolution in [Ta] works in this case, but the differential ∂ in (4.6) and (4.7) is more complicated since it uses a divided powers algebra. Theorem 4.11 can be extended to work in this case, but the resolution will have a more complicated differential. 5. Upper bounds for Betti numbers of squarefree monomial ideals In this section we study Betti numbers of squarefree monomial ideals in Clements–Lindström rings of the form W = S /(xa1 , xa2 , . . . , xna ). Order the variables by x1 , . . . , xn and consider the lex order lex in W . A squarefree monomial ideal L in W is called squarefree lex if the following property holds: if m ∈ L is a squarefree monomial and m lex m is a squarefree monomial of the same degree then m ∈ L. For a squarefree monomial ideal M in W , the squarefree Hilbert function sHilb( M , p ) : N → N of M is deﬁned by sHilb( M , p ) = the number of squarefree monomials of degree p in M . Lemma 5.1. Let a 2 be a positive integer and W = S /(xa1 , xa2 , . . . , xna ). Let M and M be squarefree monomial ideals in W . Then M and M have the same Hilbert function if and only if M and M have the same squarefree Hilbert function. Proof. For any squarefree monomial ideal M in W , we have the decomposition of kvector spaces M= m k xi : i ∈ {m} / xai −1 : i ∈ {m} , m∈ M where m ranges over the squarefree monomials in M. Thus n (dimk M p )t p = p 0 sHilb( M , p )t p 1 + t + · · · + t a−2 p =0 The statement follows from the above formula. 2 p . 54 V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 Note that Lemma 5.1 does not hold if a1 , a2 , . . . , an are not equal. For example, the ideals (x1 x2 ) and (x2 x3 ) in W = S /(x21 , x32 , x43 ) have the same squarefree Hilbert function, but do not have the same Hilbert function. Corollary 5.2. Let a 2 be an integer. For each squarefree monomial ideal M in W = S /(xa1 , xa2 , . . . , xna ) there exists a squarefree lex ideal L in W with the same Hilbert function. a a a Proof. For a squarefree monomial ideal M in S /(x11 , x22 , . . . , xnn ), where 2 a1 · · · an , the squarefree Hilbert function of M is equal to the Hilbert function of the image of M in S /(x21 , x22 , . . . , xn2 ). Thus, by Theorem 3.5, for any squarefree monomial ideal M in W , there exists a squarefree lex ideal L in W with the same squarefree Hilbert function as M. Moreover, if a1 = · · · = an then M and L have the same Hilbert function by Lemma 5.1. 2 Note that the squarefree lex ideal L in Corollary 5.2 is uniquely determined from the Hilbert function. The following result was proved by Aramova, Herzog and Hibi ([AHH, Theorem 4.4] and [AHH3, Theorem 2.9]). Theorem 5.3. Let W = S or W = S /(x21 , x22 , . . . , xn2 ). Let M be a squarefree monomial ideal in W and L the W squarefree lex ideal in W with the same Hilbert function as M. Then b iW , j ( W / M ) b i , j ( W / L ) for all i, j. We generalize the above theorem as follows. Theorem 5.4. Suppose char(k) = 0. Let a 2 be an integer and W = S /(xa1 , xa2 , . . . , xna ). Let M be a squarefree monomial ideal in W and L the squarefree lex ideal in W with the same Hilbert function as M. Then W b iW , j ( W / M ) b i , j ( W / L ) for all i, j. Proof. Let M and L be ideals in S with the same generators as M and L respectively. As we saw in Remark 3.4, by [GHP, Corollary 2.11], we get the coeﬃcientwise inequality of the graded Poincarè series W PW / M (t , u ) 1 + b iS, j S / M 1+ j ti u j 1 + t 2 ua i , j 1 b iS, j 1 + tua−1 S /L 1 + tua−1 j ti u j , 1 + t 2 ua i , j 1 where we use Theorem 5.3 for the second inequality. It is enough to prove that W PW / L (t , u ) b iS, j =1+ i , j 1 S /L 1 + tua−1 j 1 + t 2 ua ti u j . (5.5) Let m1 , m2 , . . . , mr be the minimal monomial generators of L. Since L is squarefree Borel, by Corollary 4.15, W PW / L (t , u ) = 1 + tu deg(ms ) 1sr (1 + tua−1 )deg(ms ) (1 + tu )ms −deg(ms ) (1 + t 2 ua ) ms  . V. Gasharov et al. / Journal of Algebra 325 (2011) 34–55 55 On the other hand, it follows from [AHH2, Corollary 2.3] that the Poincarè series of S / L is given by P SS / L (t , u ) = 1 + tu deg(ms ) (1 + tu )ms −deg(ms ) . b iS, j S / L t i u j = 1 + i , j 1 1sr a−1 Observe that the righthand side of (5.5) is equal to P SS/ L (t , ( 11+−tu )u ). Then t 2 ua 1 + tua−1 t, u 1 − t 2 ua deg(ms ) ms −deg(ms ) 1 + tua−1 1 + tua−1 =1+ tu deg(ms ) 1 + tu 1 − t 2 ua 1 − t 2 ua P SS / L 1sr tu deg(ms ) =1+ 1sr = as desired. (1 + tua−1 )deg(ms ) (1 + tu )ms −deg(ms ) (1 − t 2 ua )ms  W PW / L (t , u ) 2 Acknowledgments The second author is partially supported by JSPS Research Fellowships for Young Scientists. The third author is partially supported by NSF. 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